Removing triangles and ngons
Just for consideration, you have many T-splines around your hole which automatically makes neighboring geometry N-gons (N-gon is every face consisting of 5 or more vertices)



If you want less messy geometry you should get rid of T-splines. You can do this by dissolving unnecessary edges (some of them are marked in orange on the image below (x -> dissolve edges)). Dissolving edges (also dissolving vertices or faces) is really handy in cases when you are cleaning your own geometry. Also, you should merge some vertices.



And, most important. Next time you are doing something like this just do this:


Make sure that the shape in the middle has same edge count (number of vertices) as a hole that the hole that you are trying to connect it to. If not, fix it by dissolving verticies.





Select both and connect them via bridge edge loops (Ctrl+E -> Bridge edge loops).




Then if you want more geometry just add subdivision surface (you probably don't have that tight poly budget).



And one last thing. Don't be so afraid of triangles. It s true that quads are usually better, but sometimes you just gotta do what you gotta do.